Well, folks, I think I've solved the problem. I'll post this here incase someone else experiences this problem.
I went to my local JD dealer and they said, before you bring it in for service, try blowing out the gas line. That didn't make sense to me because the mower ran great for the first 45 min at high speed, then it would die till it cooled off. So, just to humor the dealer, I removed the fuel line from the filter intake, hooked up my air compressor and blew air through the fuel line. Lots of bubbles came up in the fuel tank. I blew it out good.... then hooked the line back up to the filter.
Gosh darned if I've cut the grass about 5 times since then.. about 2 hrs/cutting... and no engine death! Could this have really fixed my problem? Seems like that is a real possibility!